Melt blowing is a widely used process for manufacturing nonwoven fiber products with applications spanning healthcare, agriculture, transportation, and infrastructure, among others. The process includes extrusion of a polymer melt through orifices, drawing fiber using high-speed air jets, solidification by cooling with entrained ambient air, and collection in the form of a fiber mat. The structural features and properties of the final mat are determined by a complex interplay between materials selection and fiber dynamics, air flow, and temperature characteristics from die to collector. For the latter, both process variable values and geometrical factors have substantial influence. Many experimental investigations have advanced fundamental understanding in this area, but these studies are challenging due to high air velocities, high temperatures, and the often space-constrained nature of the process, especially near the die exit. Such complexities have sparked significant interest in developing mathematical models and using computer simulations to reveal deeper fundamental insights. Herein, we review advances in melt blowing simulations by presenting employed methods and key findings in the area. We finish by describing some challenges and opportunities for further research.